A regularisation of the Coleff-Herrera residue current
type: Item
instance of: scholarly article
title: A regularisation of the Coleff-Herrera residue current
zbMATH Open document ID: 1056.32007
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2004-11-23T00:00:00Z
review text: The aim of this note is to outline a proof of the following result. Let \(X\) be a complex manifold and let \(f = (f_1,f_2): X\to\mathbb C^2\) be a holomorphic mapping. Assume that \(f\) defines a complete intersection in \(X\), i.e. that \(V_f=\{f_1=f_2 = 0\}\) has codimension 2 in \(X\). Then \[ \lim_{\varepsilon_1,\varepsilon_2\to0}\int{\overline\partial}\,\frac{\overline f_1}{| f_1| ^2+\varepsilon_1}\wedge {\overline\partial}\,\frac{\overline f_2}{| f_2| ^2+ \varepsilon_2}\wedge \varphi=\left[{\overline\partial}\,\frac{1}{f_1}\wedge {\overline\partial}\,\frac{1}{f_2}\right]\cdot\varphi \] for all test forms \(\varphi\) of bidegree \((n,n-2)\). Hence, the Coleff-Herrera current can be obtained as the unrestricted (weak) limit of a smooth \((0,2)\)-form depending on two parameters.
Mathematics Subject Classification ID: 32C30
Mathematics Subject Classification ID: 32A27
zbMATH DE Number: 2118306
zbMATH Keywords: Coleff-Herrera residue current
zbMATH Keywords: residue integral
zbMATH Keywords: smooth regularisation
zbMATH Keywords: complete intersection
zbMATH Keywords: Mellin transform
author: Håkan Samuelsson Kalm
MaRDI profile type: Publication
full work available at URL: j.crma.2004.06.003
OpenAlex ID: W2000718571
cites work: Les courants residuels associes à une forme méromorphe
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cites work: A calculus for meromorphic currents.
cites work: Residue Integrals and their Mellin Transforms
cites work: Q4884258
DOI: 10.1016/J.CRMA.2004.06.003
description: scientific article; zbMATH DE number 2118306